Chapmen-Kolmogorov master equation

Since the formal chemical kinetics operates with large numbers of particles participating in reaction, they could be considered as continuous variables. However, taking into account the atomistic nature of defects, consider hereafter these numbers N as random integer variables. The chemical reaction can be treated now as the birth-death process with individual reaction events accompanied by creation and disappearance of several particles, in a line with the actual reaction scheme [16, 21, 27, 64, 65], Describing the state of a system by a vector N = TV),..., Ns, we can use the Chapmen-Kolmogorov master equation [27] for the distribution function P(N, t)... 

The master equation is an equivalent form of the Chapman-Kolmogorov equation for Markov processes, but it is easier to handle and more directly related to physical concepts. It will be the pivot of most of the work in this book. 

Consider a Markov process, which for convenience we take to be homogeneous, so that we may write Tx for the transition probability. The Chapman-Kolmogorov equation (IV.3.2) for Tx is a functional relation, which is not easy to handle in actual applications. The master equation is a more convenient version of the same equation it is a differential equation obtained by going to the limit of vanishing time difference t. For this purpose it is necessary first to ascertain how Tx> behaves as x tends to zero. In the previous section it was found that TX (y2 yl) for small x has the form ... 

This differential form of the Chapman-Kolmogorov equation is called the master equation. 

Not only is the master equation more convenient for mathematical operations than the original Chapman-Kolmogorov equation, it also has a more direct physical interpretation. The quantities W(y y ) At or Wnn> At are the probabilities for a transition during a short time At. They can therefore be computed, for a given system, by means of any available approximation method that is valid for short times. The best known one is time-dependent perturbation theory, leading to Fermi s Golden Rule f)... 

This interpretation of the master equation means that is has an entirely different role than the Chapman-Kolmogorov equation. The latter is a nonlinear equation, which results from the Markov character, but contains no specific information about any particular Markov process. In the master equation, however, one considers the transformation probabilities as given by the specific system, and then has a linear equation for the probabilities which determine the (mesoscopic) state of that system. 

As a preliminary to studying multi-state transport processes, it is advislble to at least become acquainted with the so-called "master equation , which is essentially a Chapman-Kolmogorov form specialized for time-stationary Markov (short-memory) processes (Fox, 1978 Van Kampen, 1981) ... 

With this definition, we can readily convert the Chapman-Kolmogorov equation to a phase space master equation... 

Using Eq. 13.59 we write the differential form of the Chapman-Kolmogorov equation, or the master equation of the stochastic process X(f), as follows ... 

---